The.Algorithm.Design.Manual.Springer-Verlag.1998

Voronoi Diagrams
Aurenhammer [Aur91] and Fortune [For92] provide excellent surveys on Voronoi diagrams and
associated variants such as power diagrams. The first algorithm for constructing Voronoi
diagrams was based on divide-and-conquer and is due to Shamos and Hoey [SH75]. Good expositions of
Fortune's sweepline algorithm for constructing Voronoi diagrams in [For87] include [O'R94].
Good expositions on the relationship between Delaunay triangulations and (d+1)-dimensional convex
hulls [ES86] include [O'R94].
In a kth-order Voronoi diagram, we partition the plane such that each point in a region is closest to the
same set of k sites. Using the algorithm of [ES86], the complete set of kth-order Voronoi diagrams can be
constructed in time. By doing point location on this structure, the k nearest neighbors to a query
point can be found in . Expositions on kth-order Voronoi diagrams include [O'R94, PS85].
The smallest enclosing circle problem can be solved in time using (n-1)st order Voronoi
diagrams [PS85]. In fact, there exist linear-time algorithms based on low-dimensional linear
programming [Meg83]. A linear algorithm for computing the Voronoi diagram of a convex polygon is
given by [AGSS89].
Related Problems: Nearest neighbor search (see page ), point location (see page ), triangulation
(see page ).
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Algorithms
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